Method for calbrating a pipelined  continuous-time sigma delta modulator

ABSTRACT

Traditionally, pipelined continuous-time (CT) sigma-delta modulators (SDM) have been difficult to build due at least in part to the difficulties in calibrating the pipeline. Here, however, a pipelined CT SDM is provided that has an architecture that is conducing to being calibrated. Namely, the system includes a digital filter and other features that can be adjusted to account for input imbalance errors and well as quantization leakage noise.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is continuation from U.S. patent application Ser. No.12/899,158, filed Oct. 6, 2010, which is related to co-pending U.S.patent application Ser. No. 12/899,205, entitled “PIPELINEDCONTINUOUS-TIME SIGMA DELTA MODULATOR,” filed Oct. 6, 2010, which arehereby incorporated by reference for all purposes.

TECHNICAL FIELD

The invention relates generally to data converters and, moreparticularly, to continuous-time (CT) sigma-delta modulators (SDMs) orsigma-delta analog-to-digital converters (ADCs).

BACKGROUND

Referring to FIG. 1 of the drawings, the reference numeral 100 generallydesignates a pipelined discrete-time (DT) SDM. With a DT data converter,an analog input signal (such as signal AIN) is sampled by asample-and-hold (S/H) circuit (such as S/H circuit 102) at discretepoints in time or sampling instants, and the samples are converted todigital. Here, two SDM stages 104-1 and 104-2 are used in a pipelineconfiguration to perform the conversion for each of the samples. Each ofstages 104-1 and 104-2 respectively comprise summing circuits116-1/118-1 or 116-2/118-2, a delay 120-1 or 120-2, quantizer 122-1 or122-2, digital low pass filter (LPF) 124-1 or 124-2, anddigital-to-analog converter (DAC) 128-1 or 128-2. Additionally, stage104-1 also includes digital filter 126. Between stages 104-1 and 104-2,there are several other components that enable stages 104-1 and 104-2 tooperate as a pipeline; namely, these components are delay 108, summingcircuit 110, amplifiers 112 and 114, analog LPF 113, and digital outputcircuit 106.

In operation, DT SDM 100 converts the analog input signal AIN to digitaloutput signal DOUT. To accomplish this, a sample of the analog inputsignal AIN is provided to stage 104-1 (by S/H circuit 102), where thesample is converted to digital using conventional sigma-deltamodulation. The same sample is provided to delay 108 so as to providestage 104-1 with sufficient time to perform the data conversion. Thedifference analog representation of the data conversion (from DAC 128-1)and the sampled analog input signal AIN (from delay 108) or residue isdetermined by summing circuit 110. This residue is amplified andfiltered by amplifiers 112 and 114 and analog LPF 113 and provided tostage 104-2. Stage 104-2 can then convert the residue to digital usingconventional sigma-delta modulation. The digital output circuit 106 thengenerates the digital output signal DOUT based the output from eachpipeline 104-1 and 104-2.

This architecture, however, is incompatible with CT sigma-deltamodulation. With DT sigma-delta modulation, the input to the stages(i.e., stages 104-1 and 104-2) is constant during conversion because theS/H circuit 102 holds the sampled analog input signal AIN. In contrast,an input to stages of a pipeline would be varying. Looking to DT SDM100, it specifically employs a delay 108 so that stages 104-1 and 104-2perform sigma-delta modulation on the same sample. If one were to removethe S/H circuit 102 so as to provide a continuously varying signal(i.e., analog input signal AIN) directly to stage 104-1 and delay 108,DT SDM 100 would not function.

Some other conventional circuits are: U.S. Pat. No. 5,729,230; U.S. Pat.No. 6,788,232; U.S. Pat. No. 7,460,046; U.S. Pat. No. 7,486,214.

SUMMARY

A preferred embodiment of the present invention, accordingly, providesan apparatus. The apparatus comprises a first continuous-time (CT)sigma-delta modulator (SDM) that receives an analog input signal; adigital-to-analog converter (DAC) that is coupled to the first CT SDM; afirst summing circuit that receives the analog input signal and that iscoupled to the DAC, wherein the first summing circuit determines adifference between the analog input signal and an output from the DAC;an amplifier that is coupled to the summing circuit, wherein theamplifier has a first gain, and wherein the amplifier includes a filter;a second CT SDM that is coupled to the amplifier; a digital gain circuitthat is coupled to the second CT SDM, wherein the digital gain circuithas a second gain, and wherein the second gain is substantially theinverse of the first gain, and wherein the amplifier, the second CTSDM,and the DAC collectively have a first transfer function; a digitalfilter that is coupled to the first CT SDM, wherein the digital filterhas a second transfer function, wherein the second transfer functionsubstantially matches the first transfer function; and a second summingcircuit that is coupled to the digital filter and the digital gaincircuit.

In accordance with a preferred embodiment of the present invention, theDAC further comprises a first DAC having a third gain, and wherein thedigital filter has a fourth gain, and wherein the second CT SDM furthercomprises: a third summing circuit that is coupled to the amplifier; anSDM filter that is coupled to the third summing circuit; a quantizerthat is coupled to the SDM filter; and a second DAC that is coupled tothe quantizer and the third summing circuit, wherein the third summingcircuit determines a difference between an output of the amplifier andan output of the second DAC, and wherein the second DAC has a fifthgain, and wherein the ratio of the third gain to the fifth gain isapproximately equal to the fourth gain.

In accordance with a preferred embodiment of the present invention, theSDM filter and the quantizer further comprise a first SDM filter and afirst quantizer, and wherein the first CT SDM further comprises: afourth summing circuit that receives the analog input signal; a secondSDM filter that is coupled to the fourth summing circuit; a secondquantizer that is coupled to the second SDM filter; and a second DACthat is coupled to the second quantizer and the fourth summing circuit,wherein the fourth summing circuit determines a difference between theanalog input signal and an output of the second DAC.

In accordance with a preferred embodiment of the present invention, theapparatus further comprises: an analog delay line that receives theanalog input signal is coupled to the first summing circuit; and adigital predictor that is coupled between the first CT SDM and the firstDAC.

In accordance with a preferred embodiment of the present invention, theapparatus further comprises an analog predictor that receives the analoginput signal and that is coupled to the fourth summing circuit.

In accordance with a preferred embodiment of the present invention, theamplifier further comprises a first amplifier, and wherein the apparatusfurther comprises a second amplifier that is coupled to the firstsumming circuit and that receives the analog input signal.

In accordance with a preferred embodiment of the present invention, thesecond amplifier has a third gain, and wherein the third gain isdimensioned to minimize an autocorrelation of an output of the second CTSDM.

In accordance with a preferred embodiment of the present invention, theapparatus further comprises an output circuit that is coupled to thesecond circuit and that provides a digital output signal.

In accordance with a preferred embodiment of the present invention, anapparatus is provided. The apparatus comprises an input terminal; afirst stage of a pipeline including: a first CT SDM that is coupled tothe input terminal; and a digital filter that is coupled to the first CTSDM, wherein the digital filter has a first transfer function; a secondstage of a pipeline including: a first summing circuit that is coupledto the input terminal, wherein the first summing circuit is adapted todetermine a difference; an amplifier that is coupled to the firstsumming circuit, wherein the amplifier has a first gain, and wherein theamplifier includes a filter; a second CT SDM that is coupled to thefirst amplifier; and a digital gain circuit that is coupled to thesecond CT SDM, wherein the digital gain circuit has a second gain thatis an inverse of the first gain; a DAC that is coupled between the firstCT SDM and the first summing circuit, wherein the amplifier, the DAC,and the second CT SDM collectively have a second transfer function; anda second summing circuit that is coupled to each stage of the pipeline,wherein the first transfer function is adjusted to substantially matchthe second transfer function.

In accordance with a preferred embodiment of the present invention, theDAC further comprises a first DAC having a third gain, and wherein thedigital filter has a fourth gain, and wherein the second CT SDM furthercomprises: a third summing circuit that is coupled to the amplifier; anSDM filter that is coupled to the third summing circuit; a quantizerthat is coupled to the SDM filter; and a second DAC that is coupled tothe quantizer and the third summing circuit, wherein the third summingcircuit determines a difference between an output of the amplifier andan output of the second DAC, and wherein the second DAC has a fifthgain, and wherein the ratio of the third gain to the fifth gain isapproximately equal to the fourth gain.

In accordance with a preferred embodiment of the present invention, theSDM filter and the quantizer further comprise a first SDM filter and afirst quantizer, and wherein the first CT SDM further comprises: afourth summing circuit that receives the analog input signal; a secondSDM filter that is coupled to the fourth summing circuit; a secondquantizer that is coupled to the second SDM filter; and a second DACthat is coupled to the second quantizer and the fourth summing circuit,wherein the fourth summing circuit determines a difference between theanalog input signal and an output of the second DAC.

In accordance with a preferred embodiment of the present invention, theamplifier further comprises a first amplifier, and wherein the apparatusfurther comprises a second amplifier that is coupled to the firstsumming circuit and that receives the analog input signal.

In accordance with a preferred embodiment of the present invention, thesecond amplifier has a third gain, and wherein the third gain isadjusted by the controller to minimize an autocorrelation of an outputof the second CT SDM.

In accordance with a preferred embodiment of the present invention, theapparatus further comprises an output circuit that is coupled to thesecond circuit and that provides a digital output signal.

In accordance with a preferred embodiment of the present invention, anapparatus is provided. The apparatus comprises an input terminal thatreceives an analog input signal; a first stage of a pipeline including:a first CT SDM including: a first summing circuit that is coupled to theinput terminal so as to receive the analog input signal; a first SDMfilter that is coupled to the first summing circuit; a first quantizerthat is coupled to the first SDM filter; and a first DAC that is coupledto the first quantizer and the first summing circuit, wherein the firstsumming circuit determines a difference between the analog input signaland an output of the second DAC; and a digital filter that is coupled tothe first CT SDM, wherein the digital filter has a first transferfunction; a second stage of a pipeline including: a first amplifier thatis coupled to the input terminal so as to receive the analog inputsignal, wherein the first amplifier has a first gain; a second summingcircuit that is coupled to the first amplifier, wherein the secondsumming circuit is adapted to determine a difference; a second amplifierthat is coupled to the second summing circuit, wherein the secondamplifier has a second gain, wherein the second amplifier includes afilter; a second CT SDM having: a third summing circuit that is coupledto the second amplifier; a second SDM filter that is coupled to thethird summing circuit; a second quantizer that is coupled to the secondSDM filter; and a second DAC that is coupled to the second quantizer andthe third summing circuit, wherein the third summing circuit determinesa difference between an output of the second amplifier and an output ofthe second DAC; and a digital gain circuit that is coupled to the secondCT SDM, wherein the third amplifier has a third gain that is an inverseof the second gain; a third DAC that is coupled between the first CT SDMand the second summing circuit, wherein the third DAC, the second CTSDM, and the second amplifier collectively have a second transferfunction; a fourth summing circuit that is coupled to each stage of thepipeline, wherein the first transfer function is adjusted tosubstantially match the second transfer function, and wherein the firstgain is adjusted to minimize an autocorrelation of an output of thesecond CT SDM, and wherein a gain of the digital filter to beapproximately equal to a ratio of the gains of the second and thirdDACs; and an output circuit that is coupled to the fourth summingcircuit and that provides a digital output signal.

In accordance with a preferred embodiment of the present invention, amethod for calibrating at least a portion of a pipelined continuous-time(CT) sigma-delta modulator (SDM) is provided, The CT SDM includes afirst stage, a second stage, and a first digital-to-analog converter(DAC) coupled between the first and second stages, and a digital filterthat is coupled to the first and second stages, and wherein the secondstage includes a second DAC. The method comprises determining a ratio ofa gain of the first DAC to a gain of the second DAC; adjusting a gain ofa digital filter to be approximately equal to the ratio of the gain ofthe first DAC to the gain of the second DAC; and adjusting the digitalfilter to maximize a cross-correlation between an output of the digitalfilter and the output of the second stage.

In accordance with a preferred embodiment of the present invention, themethod further comprises: disabling the first DAC, wherein the first DACis located between a first stage and a second stage of the pipelined CTSDM; applying a predetermined input signal to the second stage while thefirst DAC is disabled; enabling the first DAC; disabling the second DACwithin the second stage; and applying the predetermined input signal tothe second stage while the using the first DAC as a feedback DAC for thesecond stage.

In accordance with a preferred embodiment of the present invention, themethod further comprises determining a gain of an amplifier located inthe second stage that minimizes an autocorrelation of an output of thesecond stage.

In accordance with a preferred embodiment of the present invention, amethod for calibrating at least a portion of a pipelined CT SDM isprovided. The method comprises disabling a first DAC, wherein the firstDAC is located between the first stage and the second stage of thepipelined CT SDM; applying a predetermined input signal to the secondstage while the first DAC is disabled; enabling the first DAC; disablinga second DAC within the second stage; applying the predetermined inputsignal to the second stage while the using the first DAC as a feedbackDAC for the second stage; determining gains of the first and secondDACs; and adjusting a gain of a digital filter to be a function of thegains of the first and second DACs.

In accordance with a preferred embodiment of the present invention, thedigital filter is coupled to the first and second stages.

In accordance with a preferred embodiment of the present invention, thefunction is a ratio of the gains of the first and second DACs.

In accordance with a preferred embodiment of the present invention, themethod further comprises determining a gain of an amplifier located inthe second stage that minimizes an autocorrelation of an output of thesecond stage.

In accordance with a preferred embodiment of the present invention, themethod further comprises adjusting the digital filter to maximize across-correlation between an output of the digital filter and the outputof the second stage.

The foregoing has outlined rather broadly the features and technicaladvantages of the present invention in order that the detaileddescription of the invention that follows may be better understood.Additional features and advantages of the invention will be describedhereinafter which form the subject of the claims of the invention. Itshould be appreciated by those skilled in the art that the conceptionand the specific embodiment disclosed may be readily utilized as a basisfor modifying or designing other structures for carrying out the samepurposes of the present invention. It should also be realized by thoseskilled in the art that such equivalent constructions do not depart fromthe spirit and scope of the invention as set forth in the appendedclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention, and theadvantages thereof, reference is now made to the following descriptionstaken in conjunction with the accompanying drawings, in which:

FIG. 1 is a block diagram of a conventional pipelined DT SDM;

FIG. 2 is a block diagram of an example of a pipelined CT SDM inaccordance with a preferred embodiment of the present invention; and

FIGS. 3 and 4 are block diagrams of examples of the sub-CT SDMs of FIG.2.

DETAILED DESCRIPTION

Refer now to the drawings wherein depicted elements are, for the sake ofclarity, not necessarily shown to scale and wherein like or similarelements are designated by the same reference numeral through theseveral views.

Turning to FIGS. 2-4, an example of a pipelined CT SDM 200 in accordancewith a preferred embodiment of the present invention can be seen. As anexample, CT SDM 200 is a two-stage pipeline; however, CT SDM 200 can bescaled to include more stages. In this example, CT SDM 200 generallycomprises CT SDMs (or sub-CT SDMs) 202 and 212, DAC 204, digital filter206, amplifiers 220 and 210, digital gain circuit 214, summing circuits208 and 216, output circuit 218, and an adjustable delay 222. CT SDM 202(which can be seen in FIG. 3) generally comprises summing circuit 302,SDM filter 304, quantizer 306, and DAC 308, and CT SDM 212 generallycomprises summing circuit 402, SDM filter 404, quantizer 405, and DAC408. Also, CT SDM 202 can be a lower order modulator (i.e., order of 1or 2), while CT SDM 212 can be a higher order modulator (i.e., ordergreater than 3) with aggressive noise shaping. Moreover, amplifier 210also includes a filter. Amplifiers 220 and 210 can take on many forms,including but not limited to voltage-to-voltage amplifiers (i.e.,operational amplifiers), voltage-to-current amplifiers with a currentgain (i.e, transconductance amplifiers or variable resistors),current-to-voltage amplifiers (i.e., transimpedance amplifiers), orcurrent-to-current amplifiers (i.e., current mode amplifiers having atopology that depends on the input signal AIN).

In order for CT SDM to function, CT SDM 200 is generally calibrated tocompensate for system mismatches, but to make any calibrations, thesources of mismatch and error should be identified. Each of DACs 204,308, and 408 each have gains of g₄, g₁, and g₂, while amplifiers 220 and210 and digital gain circuit 214 have gains of g₃, g₅, and g₆. Gain g₆can be adjusted to be approximately equal to be the inverse of gain g₅(or g₆=1/g₅), which generally eliminates the effect of amplifiers 210.Alternatively, the gain g₅ can be 1 so as to use the filter incorporatedtherein. Additionally, each of SDM filters 304 and 306 include a gain of1/g₁ and 1/g₂, respectively, to compensate for the gains of theirrespective DACs 308 and 408, and digital filter 206 has a gain of g_(F)and a transfer function of C_(F)(z). As a result, the output Y₁(z)(which is in the frequency domain or z-domain) from CT SDMs 202 is

$\begin{matrix}{{{Y_{1}(z)} = {\frac{{S_{1}(z)}{X(z)}}{g_{1}} + {{N_{1}(z)}{Q_{1}(z)}}}},} & (1)\end{matrix}$

where, S₁(z) is the Signal Transfer Function (STF) and N₁(z) is theNoise Transfer Function (NTF) of an equivalent Discrete-Time Sigma DeltaModulator to which the CT SDM 200 is mapped for purposes of analysisusing techniques well known to those skilled in the art, Q₁(z) is thequantization error of the quantizer and X(z) is the discrete timeequivalent of the continuous-time input. This leads to the outputY_(1,N)(z) from digital filter 206 being:

$\begin{matrix}{{{Y_{1,N}(z)} = {{{Y_{1}(z)}g_{F}{C_{F}(z)}} = {{\left( \frac{g_{F}}{g_{1}} \right){S_{1}(z)}{X(z)}{C_{F}(z)}} + {g_{F}{N_{1}(z)}{Q_{1}(z)}{C_{F}(z)}}}}},} & (2)\end{matrix}$

The input R(z) into CT SDM 212 is a combination of the output fromamplifier 220 and output Y₁(z) yielding:

$\begin{matrix}{{R(z)} = {{{g_{3}{X(z)}} - {g_{4}{Y_{1}(z)}}} = {{g_{3}{X(z)}} - {\left( \frac{g_{4}}{g_{1}} \right){S_{1}(z)}{X(z)}} - {g_{4}{N_{1}(z)}{{Q_{1}(z)}.}}}}} & (3)\end{matrix}$

Now using the same rationale applied to CT SDM 202 (because thestructures of CT SDMs 202 and 212 are similar), the output Y₂(z) for CTSDM 212 is:

$\begin{matrix}{{Y_{2}(z)} = {{\frac{{S_{2}(z)}{R(z)}}{g_{2}} + {{N_{2}(z)}{Q_{2}(z)}}} = {{\left( \frac{g_{3}}{g_{2}} \right){S_{2}(z)}{X(z)}} - {\left( \frac{g_{4}}{g_{1}g_{2}} \right){S_{2}(z)}{S_{1}(z)}{X(z)}} - {\left( \frac{g_{4}}{g_{2}} \right){S_{2}(z)}{N_{1}(z)}{Q_{1}(z)}} + {{N_{2}(z)}{Q_{2}(z)}}}}} & (4)\end{matrix}$

Thus, the output Y(z) of CT SDM 200 should be:

$\begin{matrix}{{Y(z)} = {{Y_{1,N} + {Y_{2}(z)}} = {{\left( \frac{g_{F\;}}{g_{1}} \right){S_{1}(z)}{X(z)}{C_{F}(z)}} + {{N_{2}(z)}{Q_{2}(z)}} + {\left( \frac{g_{3}}{g_{2}} \right){S_{2}(z)}{X(z)}} - {\left( \frac{g_{4}}{g_{1}g_{2}} \right){S_{2}(z)}{S_{1}(z)}{X(z)}} + {g_{F}{N_{1}(z)}{Q_{1}(z)}{C_{F}(z)}} - {\left( \frac{g_{4}}{g_{2}} \right){S_{2}(z)}{N_{1}(z)}{Q_{1}(z)}}}}} & (5)\end{matrix}$

Equation (5) can then be reduced as follows:

$\begin{matrix}{{Y(z)} = {{\left( \frac{g_{F}}{g_{1}} \right){S_{1}(z)}{X(z)}{C_{F}(z)}} + {{N_{2}(z)}{Q_{2}(z)}} + {\left( \frac{{g_{1}g_{3}} - {g_{4}{S_{1}(z)}}}{g_{1}g_{2}} \right){S_{2}(z)}{X(z)}} + {\left( {{g_{F}{C_{F}(z)}} - {\left( \frac{g_{4}}{g_{2}} \right){S_{2}(z)}}} \right){N_{1}(z)}{Q_{1}(z)}}}} & (6)\end{matrix}$

Therefore, it can be easily observed that output Y(z) is a combinationof the desired output Y_(DES)(z), the input phase imbalance Y_(PI)(z),and the quantization noise leakage Y_(QNL)(z), which are as follows:

$\begin{matrix}{{{Y_{DES}(z)} = {{\left( \frac{g_{F}}{g_{1}} \right){S_{1}(z)}{X(z)}{C_{F}(z)}} + {{N_{2}(z)}{Q_{2}(z)}}}}{{Y_{PI}(z)} = {\left( \frac{{g_{1}g_{3}} - {g_{4}{S_{1}(z)}}}{g_{1}g_{2}} \right){S_{2}(z)}{X(z)}}}{{Y_{QNL}(z)} = {\left( {{g_{F}{C_{F}(z)}} - {\left( \frac{g_{4}}{g_{2}} \right){S_{2}(z)}}} \right){N_{1}(z)}{Q_{1}(z)}}}} & (7)\end{matrix}$

Looking first to the quantization noise leakage Y_(QNL)(z), this erroris related to the gains g₄, g₂, and g_(F) and transfer functionsC_(F)(z) and S₂(z). If one were to set the ratio of gains g₄ and g₂ tobe approximately equal to gain g_(F)

$\; {\left( {\frac{g_{4}}{g_{2}} = g_{F}} \right),}$

then a matching of the transfer functions C_(F)(z) and S₂(z) wouldresult in elimination of this quantization noise leakage Y_(QNL)(z).Since gain g_(F) and C_(F)(z) transfer function is adjustable (as beingpart of digital filter 206), adjustment can be based on determinationsof the gains g₄ and g₂ and transfer function.

To determine the gains g₄ and g₂, DACs 204 and 408 can be selectivelydeactivated. Initially, a test signal (of any magnitude) can be appliedto the CT SDM 200 with DAC 204 in a deactivated state and the gain g₃set to 1 so that the output Y₂(z) of CT SDM 212 can be measured. Underthese circumstances, the gain g₄ is effectively 0, allowing equation (4)to be reduced to become output Y_(2C1)(z) as follows:

$\begin{matrix}{{Y_{2C\; 1}(z)} = {{\left( \frac{1}{g_{2}} \right){S_{2}(z)}{X(z)}} + {{N_{2}(z)}{Q_{2}(z)}}}} & (8)\end{matrix}$

Then, the same test signal can be applied to CT SDM 212 with DAC 408 ina deactivated state and with DAC 204 as a feedback DAC for CT SDM 212.This changes the output Y₂(z) to become output Y_(2C2)(z) as follows:

$\begin{matrix}{{Y_{2C\; 2}(z)} = {{\left( \frac{1}{g_{4}} \right){S_{2}(z)}{X(z)}} + {{N_{2}(z)}{Q_{2}(z)}}}} & (9)\end{matrix}$

Each of outputs Y_(2C1)(z) and Y_(2C2)(z) can be measured. By dividingthe outputs Y_(2C1)(z) and Y_(2C2)(z) and noting that for a smallband-width around the signal of interest the term N₂(z)Q₂(z) isnegligible yields:

$\begin{matrix}{\frac{Y_{2C\; 1}(z)}{Y_{2C\; 2}(z)} = \frac{g_{4}}{g_{2}}} & (10)\end{matrix}$

Thus, a simple analysis of the system (which depends on the structuresof the SDM filter 404) can yield the ratio

$\frac{g_{4}}{g_{2}}.$

Typically, CT SDM 212 can be a higher order modulator (i.e., greaterthan 3) so the SDM filter 404 be, accordingly, a higher order filter.Gain g_(F) can then be adjusted to be proximately equal to the ratio

$\frac{g_{4}}{g_{2}}.$

With gain g_(F) set, the transfer function C_(F)(z) can be adjusted tosubstantially match the transfer function S₂(z). To do this, an errorfunction E that is a cross-correlation of an output Y_(1,N)(z) ofdigital filter and output Y₂(z) of CT SDM 212 is used, where errorfunction E is as follows:

E{Y _(1,N)(k),Y ₂(k)}=(Y _(1,N) *Y ₂)(k)=Σ Y _(1,N)(i)Y ₂(i+k)  (11)

This error function E is maximized when the transfer functions C_(F)(z)and S₂(z) are matched. Thus, digital filter 206 can be adjusted untilthe error function E is substantially maximized. Additionally, becausethe Q₁(z) are common terms between outputs Y_(1,N)(z) and Y₂(z), digitalfilter 206 can be blindly adjusted or calibrated.

Now, turning to the gain imbalance, the output Y₂(z) is generallycomprised shaped of Q-noise Y_(2Q)(z) and phase/gain imbalanceY_(2P1)(z), which are as follows:

$\begin{matrix}{{{Y_{2{PI}}(z)} = {{X(z)}\left( {g_{3} - {\left( \frac{g_{4}}{g_{1}} \right){S_{1}(z)}}} \right)\left( \frac{S_{2}(z)}{g_{2}} \right)}}{{Y_{2Q}(z)} = {{{N_{2}(z)}{Q_{2}(z)}} - {\left( \frac{g_{4}}{g_{2}} \right){S_{2}(z)}{N_{1}(z)}{Q_{1}(z)}}}}} & (12)\end{matrix}$

Because there can be a delay associated with amplifier 220 and DAC 204,gains g₃ and g₄ can be represented as g₃A_(d)(z) and g₄D_(d)(z), andfrom equation (11) above, it is clear that the following conditionshould substantially eliminate the gain imbalance Y_(2P1)(z):

$\begin{matrix}{{{g_{3}{A_{d}(z)}} - {\left( \frac{g_{4}}{g_{1}} \right){D_{d}(z)}{S_{1}(z)}}} = 0} & (13)\end{matrix}$

This would mean that the gain imbalance of Y_(2P1)(z) would besubstantially eliminated when the when the autocorrelation of Y₂(z)(with gains g₃ and g₄ represented as g₃A_(d)(z) and g₄D_(d)(z)) isapproximately equal to zero. Thus, by adjusting delay 222 and the gainof 220, the gain imbalance Y_(2P1)(z) cab be substantially eliminated.

Alternatively, the ratio

$\frac{g_{4}}{g_{1}}$

can be determined by selectively deactivating DACs 204 and 408, similarto the method described above to determine the ratio

$\frac{g_{4}}{g_{2}}.$

Initially, a test signal (of any magnitude) can be applied to the CT SDM200 with DAC 204 in a deactivated state and the gain g₃ set to the ratio

$\frac{g_{4}}{g_{1}}$

so that the output Y₂(z) of CT SDM 212 can be measured. Under thesecircumstances, equation (4) can be reduced to become output Y_(2D1)(z)as follows:

$\begin{matrix}{{Y_{2D\; 1}(z)} = {{\left( \frac{1}{g_{1}} \right){S_{2}(z)}{X(z)}} + {{N_{2}(z)}{Q_{2}(z)}}}} & (14)\end{matrix}$

Then, the same test signal can be applied to CT SDM 212 with DAC 408 ina deactivated state and with DAC 204 as a feedback DAC for CT SDM 212.This changes the output Y₂(z) to become output Y_(2C2)(z) as denoted inequation (9) above. Each of outputs Y_(2D1)(z) and Y_(2C2)(z) can bemeasured. By dividing the outputs Y_(2D1)(z) and Y_(2C2)(z) and notingthat, for a small band-width around the signal of interest, the termN₂(z)Q₂(z) is negligible yields:

$\begin{matrix}{\frac{Y_{2D\; 1}(z)}{Y_{2C\; 2}(z)} = \frac{g_{4}}{g_{1}}} & (15)\end{matrix}$

Thus, a simple analysis of the system (which depends on the structuresof the SDM filter 404) can yield the ratio

$\frac{g_{4}}{g_{1}}.$

Thus, by adjusting gain g₃ to be approximately equal to the ratio

$\frac{g_{4}}{g_{1}}$

with this foreground calibration scheme (as opposed to the backgroundscheme described above), gain imbalance can be substantially eliminated.

To generally eliminate phase imbalance, either a digital predictor 220or digital predictor with an analog delay line 222 can be employed. Thetuning of either the digital predictor 220 or the analog delay line 222can be done by minimizing the autocorrelation (similar to the schemedescribed above). For the digital predictor 220, for example, an analogdelay line 222 can be introduced such that the delay through delay line222 is greater than that through the CT SDM 202 so as to allow digitalpredictor 220 to be tuned such that the auto-correlation is minimized.

Having thus described the present invention by reference to certain ofits preferred embodiments, it is noted that the embodiments disclosedare illustrative rather than limiting in nature and that a wide range ofvariations, modifications, changes, and substitutions are contemplatedin the foregoing disclosure and, in some instances, some features of thepresent invention may be employed without a corresponding use of theother features. Accordingly, it is appropriate that the appended claimsbe construed broadly and in a manner consistent with the scope of theinvention.

1. A method for calibrating at least a portion of a pipelinedcontinuous-time (CT) sigma-delta modulator (SDM), wherein the CT SDMincludes a first stage, a second stage, and a first digital-to-analogconverter (DAC) coupled between the first and second stages, and adigital filter that is coupled to the first and second stages, andwherein the second stage includes a second DAC, the method comprising:determining a ratio of a gain of the first DAC to a gain of the secondDAC; adjusting a gain of a digital filter to be approximately equal tothe ratio of the gain of the first DAC to the gain of the second DAC;and adjusting the digital filter to maximize a cross-correlation betweenan output of the digital filter and the output of the second stage. 2.The method of claim 1, wherein the method further comprises: disablingthe first DAC, wherein the first DAC is located between the first stageand the second stage of the pipelined CT SDM; applying a predeterminedinput signal to the second stage while the first DAC is disabled;enabling the first DAC; disabling the second DAC within the secondstage; and applying the predetermined input signal to the second stagewhile the using the first DAC as a feedback DAC for the second stage. 3.The method of claim 2, wherein the method further comprises determininga gain of an amplifier located in the second stage that minimizes anautocorrelation of an output of the second stage.
 4. A method forcalibrating at least a portion of a pipelined CT SDM, the methodcomprising: disabling a first DAC, wherein the first DAC is locatedbetween a first stage and a second stage of the pipelined CT SDM;applying a predetermined input signal to the second stage while thefirst DAC is disabled; enabling the first DAC; disabling a second DACwithin the second stage; applying the predetermined input signal to thesecond stage while the using the first DAC as a feedback DAC for thesecond stage; determining gains of the first and second DACs; andadjusting a gain of a digital filter to be a function of the gains ofthe first and second DACs.
 5. The method of claim 4, wherein the digitalfilter is coupled to the first and second stages.
 6. The method of claim5, wherein the function is a ratio of the gains of the first and secondDACs.
 7. The method of claim 6, wherein the method further comprisesdetermining a gain of an amplifier located in the second stage thatminimizes an autocorrelation of an output of the second stage.
 8. Themethod of claim 7, wherein the method further comprises adjusting thedigital filter to maximize a cross-correlation between an output of thedigital filter and the output of the second stage.